We use the lace expansion to analyse networks of mutually-avoiding
self-avoiding walks, having the topology of a graph. The networks are
defined in terms of spread-out self-avoiding walks that are permitted
to take large steps. We study the asymptotic behaviour of networks in
the limit of widely separated network branch points, and prove
Gaussian behaviour for sufficiently spread-out networks on
$\mathbb{Z}^d$ in dimensions $d>4$.